AnalysesZDM 99/2 from. This atom is symbolised on earth by the seed of the Digitaria Exilis (Palau Marti 1957: 53–55). There are two main versions of the creation. Both refer to Amma, the supreme Lord the heavenly regions, and to the appearance of sets of twins who became the ancestors of the Dogon people. Regardless of whether Amma married the earth and made it fertile with his divine seed (rain), or man developed from the egg of the world, the principle of duality always appears throught the first set of twins, Nommo, (one male and the other female). This duality is reinforced by the two collar bones from which the embryo of man developed, (where duality is translated as axial symmetry) and by the two complementary “halfplacentas” from the egg of the world or primordial matrix. Double duality is immediately apparent as each of these “half-placentas” produces a hermaphrodite couple. These couples became the 4 mythical ancestors who in turn gave origin to the first 80 descendents. When dealing with the general and totalising character of 4, we can see how these 4 ancestors joined in chains of symblic correspondence, not just in 4 tribes, 4 trees and 4 social institutions, but also in the 4 cardinal points and the 4 fundamental elements. Amma used 4 elements, contained in the initial living atom, to create the substance that later became vivified through the “parole”: “la parole” being the life of the world, born of water (Dieterlen 1982: 18). Earth was sullied by the introduction of trouble by Nommo’s rival, Ogo (also known as Yurugu or the Pale Fox) who came down to earth after appropriating the 8 seeds initially created by Amma and after having stolen the secret of “la parole” (Dieterlen 1982: 17; Beaudoin 1984: 70). Curiously, Ogo’s revolt seems to have been caused by his particular form of birth: he was born a male from one half of the original egg, but without a female twin who would have integrated him into the principle of normal duality. The world was re-purified by Amma through the death and resurrection of Nommo. Nommo’s sacrificed body was divided and thrown in the 4 directions of space. The development of the original mythical duality explains the symbolic importance of 8 by multiplicative unfolding, and intrinsically joins the semantics of spatial representations that is an ever-present in Dogon thought. An analysis of the symbolic dimension of 3 N 3 N The territory of the Dogon is situated between the Mali city of Mopti and the border of Burkina Faso. which turns into the egg of the world (aduno talu) where the cosmos comes A presentation to the First International Conference on Ethno- I 266 mathematics. Lisboa (Portugal) Abstract: This paper deals with the following aspects of the culture of the Dogons (Mali): the specific mythological context and the related cognitive system. Dioula) and the Sudanese (Dogon. the spiral as a coreography of thought. Spain. die Spirale als Choreographie des Denkens. However. Der Beitrag behandelt folgende Aspekte der Kultur der Dogonen (Mali): der spezifische mythologische Kontext und das zugehörige kognitive System. just known by certain initiated persons and used in specific ritual ceremonies. Bozo and Peuhl. fundamental poles in the Dogon numerical symbology. 1998 The Dogon have 266 signs that prefigure the categories of everything in the universe. Kurzreferat: Ethnomathematik und symbolisches Denken: Die Kultur der Dogonen. both objects and beings: “these signs. die Bedeutung der 5. the density of 5’s significance. Among the sedentary population there are two main groups: the Mandingue (Bambara. 66 . ethnomathematics “logosymbols” as “event” and social meaning. 2. Saracollé. They have a second kind of language. we are still far from appreciating the secret wealth of their hidden knowledge. Dogon mythology is based on a dualised unity – or an original uni-duality – represented by an infinitesimally small form of living atom (kize uzi). will form part of a separate study. these categories in their turn. agility and resistance. whether linked or opposed. Furthermore. namely). educational implications (transcultural expression of thought and feeling). A cognitive system based on a principle of exponential duality Although Dogon thought has been researched for many years. After the dryness of 1970. a secret and rhythmics language (Sigi So). University of Granada. The cliffs of Bandiagara are a main place of Dogon’s life. substitutes and images constitute a vast system of correspondances. grundlegende Pole der numerischen Symbolik der Dogonen. vivacity. are themselves correlated” (Dieterlen 1982: 69. but their daily tongue is radically different from those that are spoken by their neighbours. Some fundamental poles in the complex Dogon numerical symbology Ethnomathematics and Symbolic Thought The Culture of the Dogon Teresa Vergani. the Dogon can understand quite well several tongues (Bambara. The dogon Mali has different kinds of ethnic groups. Malinké. Peuhl and Moor) decreased significantly. in which every term is interlocked within what seem to be specific categories. Usually. but which also play a major part in Dogon symbolic thinking. ZDM-Classification: A30 1. its nomad population (Touareg. attention must be paid to the set of numerical entities that do not derive from the principle of duality. Palau Marti 1957: 56). 3. The Dogon are traditionally characterized by their sharpness. the most perfect of which are only reached by some exceptionally gifted individuals. September 2–5. Bonzo. Sonrai). Implikationen (transkultureller Ausdruck des Denkens und Fühlens). among the Dogon this knowledge has been appearing progressively in different stages of initiation. ethnomathematische “Logosymbole” als “Ereignis” und ihre soziale Bedeutung. The 60 year gap between these cyclical festivities corresponds to the time taken by some heavenly bodies to orbit around Sirius. or what might be termed the human face of time. Nommo’s rival. The fifth attribute. ages and generations. but 7 is also the number of segments that prefigure man in the egg of the world (Griaule/Dieterlen 1955: 85. After he had stolen the initial seeds created by Amma and appropriated the secret of “la parole”. which involves the invocation of the 22 members of the first mythical generations (Griaule/Dieterlen 1955: 105). 93. Ogo’s rebellion stained earth. 201–203). namely. 67 60 is also “involved in the diagram of the world” (Griaule/Dieterlen 1955: 103) yet it also played a part in Amma’s purification of earth. The Dogon week has 5 days and local market are held every 5 days. maturity and old age (Palau Marti 1957: 49). where 60 urns are on display. disorder and death (Palau Marti 1957: 53). For example. 69: Griaule/Dieterlen 1965: 86–87). which ran for 5 days when he was sacrificed. the Dogon acknowledge 22 leading families of living beings and attribute 22 joints to Nommo’s resurrected body (Dieterlen 1982: 21. It was the seventh ancestor who taught man the “word of Nommo” (Palau Marti 1957: 54). as Nommo’s sacrificed body was divided into 60 parts (Dieterlen 1982: 18). he came down from heaven and managed to take control of a field where he dug 5 rows of holes in the damp soil. and especially at the itinerant festival of the star Sigui. This temporal division of time seems to have a symbolic relation to Nommo’s blood. in a closer analysis. These are night. The contrast death/life leads to a new set of attributes assigned to Ogo. 5 and temporal maturity The Dogon venerate the first 5 generations after Nommo. where 60 Dogon chiefs are present. every individual relates to the 5 generations around his own. For the Dogon. within the 4 fundamental Dogon lineages (Palau Marti 1957: 56). the three that preceded him. 7 taboos and 7 seeds that Yurugu stole from man. In addition. as witnesses of the exteriorised and classified world (Dieterlen 1982: 46–47). as it is believed that is was during their age that all the social and religious institutions were created. as 60 is one of the bases of the mande reckoning. all of which were associated to the fertility of the earth. 5 does not seem to form a part of the symbolically relevant numerical entities of Dogon thinking. Palau Marti 1957: 55). 61–62). more specifically in “the field of the ancestors”(Dieterlen 1982: 18. According to Paulme (1985). 60 also appears as an essential symbolic value in rain rituals and in the autumn harvest. held every 60 years and dedicated to the revelation of the “parole tissée dans l’eau” and to the appearance of death in the world. his own generation and the one that will follow. children are breast-fed for 5 consecutive seasons. Houses have 5 niches on each side of the door. Thus it is the symbol of mythical and perfect completion of time. the Dogon believe that the soul only reaches the land of the ancestors 5 years after death (Beaudoin 1984: 87. around which several different cognicultural organizational systems revolve. known as the “5 attributes of Nommo”. In addition it is associated to the cult of the ancestors. . was Amma’s wife and shared the temporal and cyclical maturity symbolised by 5 – wore a skirt made of 5 handsful of fibres at the front and at the back (Griaule 1948: 26). The future Hogon would retire to a cave for 7 days in preparation for his mission among men (Dieterlen 1982: 84). adolescence. 22 dots are represented in the “déroulement des éléments” scheme. the period from birth until the age of initiation. its implied present reveals a density of significance in a large numer of occasions.ZDM 99/2 These 266 signs are also called “the 266 parts of Amma’s body” or “the 266 things of the beginning”. at funeral ceremonies. which explains the ritual of the dead (Dama) takes place every 5 years in every village. some of which we will examine below. and the cloth is considered the “centre of the world”: 22 is linked to this knowledge of dressing. drought. 5 appears as a measure of maturity reached through the achieved or of the cycle of resurrection. fertility. The brightness of the forgotten 5 At first sight. I7 The symbolism invested in 7 appears to be linked to the origins of creation. Their names were day. The organisation and pacification of the world was characterised by 5 attributes. 5 and the purification of earth Mother Earth – according to one of the Dogon creation myth. “the science of clothes” is an account of the mechanism of creation. They belong to the initial design of the “plan of the world” – that Amma conceived of and drew on the basis of water – symbolically carried out on earth. Consequently. In addition. The itinerant festivities of the star Sigui (related to Sirius and to the appearance of death in the world) were celebrated for 5 consecutive days (Dieterlen 1982: 97). Even today. it took Amma 5 days to resuscitate Nommo. Not only are there 7 orders of reproductive vibrations. youth. sterility. There were 7 sacred plants. The symbolic efficiency of number 60 extends further to include operative efficiency. Analyses 4. life. We must not forget that a man is only considered to be completely adult until he is 22 years old. However. as the tunic of the Hogon (religious leader) is made up of 22 squares (Dieterlen 1982: 61). 90. which the Dogon call “the star of Sigui” (Dieterlen 1982: 18). order and life. for the express purpose of representing these ancestors (Beaudoin 1984: 90). This cult of 5 generations exists regardless of the individual’s lineage. Furthermore. The number 5 determines the age groups that are socially recognised by the Dogon. There were 7 animals who had been the allies of Nommo (Griaule/Dieterlen 1955: 104). I 60 I 22 22 is another central pole in Dogon numerical symbolism. 174. should be noted as the central point where the other four meet. humidity. is surrounded by 60 chiefs “located in the 4 regions of space” (Dieterlen 1982: 95). a specifically Dogon choreography of thought In both mythical thinking and in the symbolic space of the graphic teaching of the Dogon. In Dogon thinking. initially. curled up. Griaule/Dieterlen 1955: 103). The same totalising function that Dogon thinking attributes to 5. in Dogon thinking. during certain ceremonial rituals. before cutting the field off from outside by building a hibiscus hedge. The Hogon’s armlet is an iron spiral (senye) which represents “the path of Nommo’s blood” after his sacrifice (Dieterlen 1982: 71). The second reason is that this spiral movement prefigures and. This “parole de compte” is intrinsically connected to 5. Beaudoin 1984: 122). there is one called the counting word.Analyses In order to free the earth from the disorder that Ogo had brought. specifically the Milky Way. When the Hogon dies. perfectly integrates the process of creation of Sigui. the relation is clarified when we consider that Sigui literally means “the navel of the sky” and Hogon (the centre of the social body) means “umbilical cord” (Dieterlen 1982: 31). which expands in a helicoidal conic structure (Griaule/Dieterlen 1965: 163. The abstract unit 10 is. 167). The spiral. The helicoidal sign that characterizes the various isomorphic placentas. which integrates knowledge relating to numbers and to calculation. of which the spiral is the dynamic expression: it performs the explicit (un)achievement of movement produced by the force inherent in a centre that affects the surrounding space and is capable of transforming “circularity” into “spirality”. Beaudoin 1984: 203. 147–148). The ritual dances for Sigui. several types of twisted fringes or cords. based on pedagogical hermeneutics. which is metrization) is used. as has been noted. The development of this circularity is here shown in a continuous form that is open to infinity. These lines define the meandering paths of water and wind. 1965: 78). are given life by spiral movement. It is important to note that 80. are danced “en rond ou en serpentant”. whose placenta is circular. This holistic cardinality causes a unification which is also translated by an indivisible symbolic circularity. This relevance should be approached for two principal reasons. If duality (or sym. is not just indirectly linked to . that is to say. they turn into that metal (Griaule 1948: 23. This means that 80 was achieved using 5 times the fingers of two hands (Griaule/Dieterlen 1955: 104). the representation of unity” (Calame-Griaule 1965: 176. appears with special clarity in meta-unity 60. since the spiral is a specific form of 68 ZDM 99/2 movement created by the totalizing nature of the centred 4. and the movement of his collar-bones is also in a spiral. the result is @ considered “the whole and perfect number. The spiral is inscribed within the egg of the world. according to Beaudoin (1984). 80. This field. where “all knowledge lies (Griaule/Dieterlen 1950: 284–285). Dogon creation myth is profoundly marked by a movement in spiral. which is the initial base for Dogon counting. Moisture. anticipates the geometric metaphor which will be proposed. which happen at the most important Dogon festivities. ellipsoidal or sinusoidal iterations are connected to this dynamic of the spiral. the number 60. which refers back to the purification of the universe by Amma. and when they touch the earth. in one sense. Amma divided Nommo’s sacrificed body into 60 pieces. Since 60 is linked to one astral revolution around the star Sigui (the Dogon name for Sirius). with specific reference to transculturality. penetrates the earth through serpentine curves. It was here that he planted 5 seeds. analogous to the sinusoidal rhythms that govern the world and life (Griaule/Dieterlen 1965: 78. Amma makes the world turn in a spiral form. “The counting word”: the symbolic and algebraic link between 5 and the mande reckoning Among the various different categories of Dogon words. The movement of the copper spiral that moves the sun is the main base for design in weaving. mistake in his search for his female twin (or lost double soul) describes a spiral movement in the earth’s placenta (Griaule/Dieterlen 1965: 185). the ancestor Lébé Sérou created 5 new parcels of land to the West of the original field. the priest shapes the holy clay from the “field of the ancestors” into a serpent in a position of rest. The other base of mande reckoning is. exactly those of the sinuous movement of reptiles (Griaule 1948: 27. one of the fundamental bases of mande reckoning. is projected on the two-dimensional surface of the earth in the form of undulating lines. The process of segmenting the egg of the world in spiral form. as we can see in the holistic behaviour of cardinality 4. the spiral is a key geometrical word with a unique relevance. seen as the set of the fingers of two hands. The Dogon believe that during menstruation. From this we can see that 5 was involved at various levels in defense against the Fox’s attacks and in the reestablishment of peace on the purified earth. “the spiral does not turn in the woman’s womb”. The symbolic . The origins myth tells how Ogo. The stars and celestial bodies. is obtained from 8 sets of 10 elements. which he then split into 4 groups of @ elements before throwing them to the 4 winds. A particularly 5 through the multiplication @ clear explanation can be found in a legend justifying the importance of 80 as the base of calculation for various West Sudanese people: “the ancestors put down 8 cowries with 5 fingers”. First. representing the form that shapes the development of life (Griaule/Dieterlen 1955: 84. is surrounded by a spiral of incandescent red copper which gives it light and movement. It is no coincidence that the Hogon. such as those for rain and the autumn harvest. thus making her temporarily barren (Griaule 1948: 80). or the plan of universal creation. 82–83). necessary for procreation. starting from the West: Amma occupies the centre. The sun’s rays are called “copper water”. and the movement of his arms describes a threedimensional spiral in space. The sun. protected from contamination by the Pale Fox (Ogo) was called “the field of the ancestor” (Dieterlen 1982: 20). An isomorphic relation between 5 and 60 is thus discovered. such as giving numbers to the @ front pillars of houses. – Cambridge: Cambridge University Press Beaudoin. 117): “The representation. but the linking between “the concept in the mind” and “the sense-image” is intrinsic (Bronckard 1977: 57.. (1991): Dictionnaire de l’ethnologie et de l’anthropologie. as presence. water and blood. is at the same time actualization: what stands before us here and now. to the discursive understanding and objective intuition of reality”. 2) points out a convergence of psycho-conceptual directional forces. – Paris: Armand Collin Bonte. a phenomenon of a symbolic nature. 1 The segmentation of the egg of the world by the first 7 vibrations (Griaule/Dieterlen 1955: 84) or the spiral of the “déroulement des éléments” contained in the original embryo and including the 22 points considered to be the “witnesses of the exteriorised and classified word” (Dieterlen 1982: 46–47). G. through the designs in Fig. The universe is run according to its rhythm. (1985): The philosophy of symbolic formes: the phenomenology of knowledge (III). it has an intelligible “soul” which can be translated as “le soi du Logos” (Kearney 1991: 14. It is within the spiral that the word.. whose “saillance cognitive” belongs to the level of the “coagulum of sense”. These examples show how the spiral acts at the level of a cognitive model which embraces the entire project of creation conceived of by the “Designer of Things”. Nattiez 1966: 81. pleasure and poetry – a feminine word linked to the loving nature of woman. Ethnomathematics and vivid “Logosymbols” Ethnomathematics is concerned by the connections that exist between the symbol. Ethnomathematics deals with vivid “logosymbols”. A conceptual image is a consensual human fact (Barnes 1990: 225). the more clearly their nature and specific unity will be revealed to us – and it will be increasingly evident that it is ultimately one and the same fundamental achievement by which the spirit rises to (. The creation of signifiers is not only indivisible from the cognitive action. M. then sun and the stars are all formed and that the weaving of cloth and the development of seeds take place. E. This spiral movement is a geometric metaphor. – Bruxelles: Pierre Mardaga Calame-Griaule. Leach 1976: 19). This analogy between spiral movement and development of seed is revealed with particular clarity in the case of the sene seed. P. and the sacred serpent that instructs the chiefs of the people moves to its rhythm. 3) inspires a methodological dynamic approach that discloses a new world of possibilities of interaction between cognitive/cultural anthropology and mathematical education through a hermeneutical approach. At a cognitive level. Fig. This complementary relationship between cognition and recognition – or between understanding and representation – develops itself bringing symbolic thought and rational thought together in an dialogical interaction which Morin calls the uni-duality of thought. – Paris: Gallimard Cassirer. so it is no surprise that “water moves in the wavy line of the weft” (Dieterlen 1982: 97. where past and present meet. This can also give another meaning to more pragmatic or mundane matters. P. According to Calame-Griaule (1965). It is in the spiral that the 8 cardinal points. We have also seen how “the word was woven into the movement of water”. According to Cassirer (1985: 108. Morin 1986: 159). 1. The germination follows a pattern represented by the centred 5. Georgin/Wagret 1970: 465). (1977): Théories du langage: une introduction critique. a metaword that expresses holistic understanding of the world and all things. (1965): Ethnologie et langage: la parole chez les Dogon. the representation and imagery (Morin 1989: 158). J. which in normal life is expressed by the category of “the good word”. Analyses 5. the solar light and the totalising virtue of the centred 4. As an intuition of living consciousness.. The logosymbol is vivid as an intentional acting representation or a constant tension feature between event and meaning. “the weaving produces a spiral pattern from the centre of the crown to the edge. Griaule 1948: 80.. (1984): Les Dogons du Mali. the spiral is the symbolic image of deep and harmonious knowledge. it is from this category that come music. (1990): Models and interpretations: selected essays. announces itself as the emanation and manifestation of a power through the concrete uniqueness of the image.. of “relevent mystery” or of “crystal culture” (Bonté/Izard 1991: 158–60. but can also be represented by a set of concentric ovals (geometrically. – Paris: PUF Bronckart. References Barnes. 6. which symbolises the path followed by the original seed (Griaule/Dieterlen 1955: 100). a discontinuous form of the spiral) or equated to “the spider’s web” (Griaule/ Dieterlen 1965: 218. kindness. According to some authors. Izard. 278).) The more closely we follow the paths taken by the universal function of representation and recognition. A. examples of the spiral’s symbolism within the scope of the Dogons’ graphic teachings. – New Haven and London: Yale University Press 69 . related to the 8 cardinal points (Griaule 1948: 120–121).) the intuitive world view.ZDM 99/2 importance of clothing for the Dogon has already been mentioned: we can now add that the fibre of the cloth is considered to be a primary form of “parole”. the 22 main categories of living beings and the 7 initial vibrations take form. J. (. G. 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(1957): Les Dogon.pt ZDM 99/2 Vorschau auf Analysethemen der nächsten Hefte Für die Analysen der Jahrgänge 31 (1999) bis 32 (2000) sind folgende Themen geplant: – TIMSS – Computergestütztes Lösen offener Probleme im Mathematikunterricht – Mathematikdidaktische Forschung im Primarbereich – Mathematik an Hochschulen lehren und lernen – Analysis an Hochschulen – Mathematik in der Ingenieurausbildung – Theoretische Betrachtungen zu Schulbuchanalysen. Dr. (1982): Le titre d’honneur des Arou (Dogon. 70 Palau Marti. Mali). E. (1973): L’image du corps et les composantes de la personne chez les Dogon. (1976): Culture and communication: the logic by which symbols are connected. (1986): La connaissance de la connaissance: anthropologie de la connaissance (I).-C. Portugal. M.. (1948): Dieu d’Eau: entretiens avec Ogotemmêli. E. (1984): Mythologie et sacrifice. E-mail: tvergani@univ-ab. (1970): Notes sur la dialectique “fantasmeréel” et ses incidences en psychothérapie. G. 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